I don't really agree with this -- I've heard plenty of Americans say both variants. Perhaps in the commonwealth countries there is some clear-cut variant, but that's not really the case in the US, I'd say.

Shouldn't the variant 'I said that I am an Engineer' only be used if the speaker is still an engineer (at the moment of speaking)?
And if he is no longer an engineer (at the moment of speaking), shouldn't 'I said that I was an Engineer' be the only possible version?

Shouldn't the variant 'I said that I am an Engineer' only be used if the speaker is still an engineer (at the moment of speaking)?
And if he is no longer an engineer (at the moment of speaking), shouldn't 'I said that I was an Engineer' be the only possible version?

I've been taught that "I said I am" is incorrect except direct speech:
I said: 'I am...' and when you're narrating then you should use sequence of tenses rules: I said I was...
Either my teachers were wrong or people often neglect rules while speaking.
Or, maybe, the rules have changed over time.

Let me be a free man, free to travel, free to stop, free to work, free to trade where I choose, free to choose my own teachers, free to follow the religion of my fathers, free to talk, think and act for myself. - Chief Joseph, Nez Perce